The Complex Exponential and Logarithm Functions in Complex Analysis
This concept covers the complex exponential function e^z and the complex logarithm as its multi-valued inverse, within complex analysis. It establishes how the exponential maps horizontal and vertical lines to rays and circles respectively (via modulus e^x and argument y), and how the logarithm — defined as ln|z| + i(arg z + 2πim) — requires branch selection (notably the principal branch) to be single-valued, since it is inherently a multi-valued, infinitely-branched function due to the periodicity of the exponential.
The Complex Exponential and Logarithm Functions in Complex Analysis
This concept covers the complex exponential function e^z and the complex logarithm as its multi-valued inverse, within complex analysis. It establishes how the exponential maps horizontal and vertica…